Counterexamples
Counterexamples is a unit of teaching within Proof, usually two to four weeks of lessons with one assessment point.
Entity profile
Every node stores the same attribute set, which is why the hierarchy can grow without new templates.
Unit
Depth 3 in the graph
math/mathematical-reasoning/proof/counterexamples
2,235
12 topics · 360 concepts · 1,800 micro-concepts
Chapters
Every chapter below is a full entity with its own skills, objectives, misconceptions and resource set.
Introducing Counterexamples
Introducing Counterexamples is a chapter within Counterexamples: a coherent run of lessons that can be taught in one go.
Working with Counterexamples
Working with Counterexamples is a chapter within Counterexamples: a coherent run of lessons that can be taught in one go.
Counterexamples in Context
Counterexamples in Context is a chapter within Counterexamples: a coherent run of lessons that can be taught in one go.
Skills and learning objectives
Skills hang off the entity; objectives hang off the skill; activities and assessment hang off the objective.
Identify counterexamples
RememberIdentify counterexamples accurately in a familiar context.
Activities — Modelled example on the board · Paired talk task · Mini whiteboard check
Assessment — Exit ticket of four short items
Identify counterexamples independently in an unfamiliar context.
Activities — Independent worksheet · Card sort · Concept-check quiz
Assessment — Marked worksheet with a marking scheme
Explain counterexamples
UnderstandExplain counterexamples accurately in a familiar context.
Activities — Modelled example on the board · Paired talk task · Mini whiteboard check
Assessment — Exit ticket of four short items
Explain counterexamples independently in an unfamiliar context.
Activities — Independent worksheet · Card sort · Concept-check quiz
Assessment — Marked worksheet with a marking scheme
Explain the reasoning behind counterexamples using correct vocabulary.
Activities — Talk-through with a partner · Write a model answer · AI Tutor Socratic session
Assessment — Extended written response, levelled
Calculate counterexamples
ApplyCalculate counterexamples accurately in a familiar context.
Activities — Modelled example on the board · Paired talk task · Mini whiteboard check
Assessment — Exit ticket of four short items
Calculate counterexamples independently in an unfamiliar context.
Activities — Independent worksheet · Card sort · Concept-check quiz
Assessment — Marked worksheet with a marking scheme
Compare counterexamples
AnalyseCompare counterexamples accurately in a familiar context.
Activities — Modelled example on the board · Paired talk task · Mini whiteboard check
Assessment — Exit ticket of four short items
Compare counterexamples independently in an unfamiliar context.
Activities — Independent worksheet · Card sort · Concept-check quiz
Assessment — Marked worksheet with a marking scheme
Explain the reasoning behind counterexamples using correct vocabulary.
Activities — Talk-through with a partner · Write a model answer · AI Tutor Socratic session
Assessment — Extended written response, levelled
Semantic relationships
Prerequisite and related edges are what make this a graph rather than a menu.
- MatricesDomain
Misconceptions
Stored on the entity so worksheets, quizzes and lesson plans all target the same known errors.
Students treat counterexamples as a rule to memorise rather than a relationship to understand.
Correction — Anchor the rule in a visual model first, then abstract it — the model is the explanation.
Learners apply the correct method to counterexamples but ignore the context or units.
Correction — Require an answer sentence with units on every applied question.
Curriculum alignment
Curricula, boards and countries are data — adding one never changes the architecture.
COM-179.1
Mapped statement covering Counterexamples.
OCR-788.2
Mapped statement covering Counterexamples.
What comes next
Curriculum order is an edge in the graph, so every entity knows the step that follows it.
Algebraic Proof
Algebraic Proof is a unit of teaching within Proof, usually two to four weeks of lessons with one assessment point.
Proof by Exhaustion
Proof by Exhaustion is a unit of teaching within Proof, usually two to four weeks of lessons with one assessment point.
Problem Solving
Problem Solving is a curriculum strand within Mathematical Reasoning — a thread that runs across grades and deepens each year.
Difficulty ladder
The same entity, six levels of demand — easy through to olympiad and exam level.
Scaffolded first attempt with worked support.
Standard curriculum demand, unscaffolded.
Multi-step questions and unfamiliar contexts.
Stretch tasks that need reasoning, not recall.
Competition-style problems built on the same idea.
Board-styled questions with mark schemes.
Learning intent
Not every visitor wants the same thing. Each intent routes to the resource that actually does that job.
Audience views
Counterexamples in Grade 3, surfaced differently for each reader.
AI Tutor context chain
Open the tutor from here and it already knows where you are — this is the sequence it recommends.
- 01Prerequisite lessonmatrices
- 02Video explanationCounterexamples modelled end to end
- 03Interactive practiceAnswer on screen with instant checking
- 04QuizTen graded items with hints
- 05FlashcardsRetrieval on a 14-day schedule
- 06Common mistakesThe misconceptions stored on this node
- 07Harder worksheetSame concept, exam-level demand
- 08Past paper questionsBoard questions on this concept
- 09Mock testTimed, marked, with feedback
- 10Book a tutorOne-to-one on exactly this gap
Free Counterexamples resources
Every format below is a free, standalone page generated from this entity — 53 of them, all interlinked.
- Free Counterexamples lessons
- Free Counterexamples lesson plans
- Free Counterexamples explanations
- Free Counterexamples worked examples
- Free Counterexamples real life examples
- Free Counterexamples common mistakes
- Free Counterexamples video lessons
- Free Counterexamples worksheets
- Free Counterexamples printable worksheets
- Free Counterexamples interactive worksheets
- Free Counterexamples homework worksheets
- Free Counterexamples revision worksheets
- Free Counterexamples word problems
- Free Counterexamples practice tests
- Free Counterexamples flashcards
- Free Counterexamples study notes
- Free Counterexamples revision notes
- Free Counterexamples cheat sheets
- Free Counterexamples formula sheets
- Free Counterexamples mind maps
- Free Counterexamples vocabulary lists
- Free Counterexamples quizzes
- Free Counterexamples timed quizzes
- Free Counterexamples adaptive quizzes
- Free Counterexamples mcqs
- Free Counterexamples true or false questions
- Free Counterexamples fill in the blanks
- Free Counterexamples matching exercises
- Free Counterexamples short questions
- Free Counterexamples long questions
- Free Counterexamples question banks
- Free Counterexamples exam questions
- Free Counterexamples past paper questions
- Free Counterexamples solved past papers
- Free Counterexamples mock exams
- Free Counterexamples activities
- Free Counterexamples stem activities
- Free Counterexamples experiments
- Free Counterexamples games
- Free Counterexamples projects
- Free Counterexamples case studies
- Free Counterexamples homework
- Free Counterexamples assignments
- Free Counterexamples reading comprehension
- Free Counterexamples writing prompts
- Free Counterexamples spelling practice
- Free Counterexamples grammar practice
- Free Counterexamples teacher guides
- Free Counterexamples parent guides
- Free Counterexamples ai tutor sessions
- Free Counterexamples practice sessions
- Free Counterexamples progress trackers
- Free Counterexamples printables
Connected resources
30 resource types are generated for Counterexamples automatically — the graph never produces an entity without them.
Concept Explanation
The idea itself, explained from first principles.
Lesson
A taught sequence with modelling and checks for understanding.
Real Life Examples
Where the idea shows up outside school.
Common Mistakes
The misconceptions and how to correct them.
AI Tutor
Socratic one-to-one practice on this exact idea.
Quiz
Graded retrieval with hints and explanations.
Question Bank
Every question filtered to this entity.
MCQs
Multiple choice items with distractor analysis.
True or False
Fast misconception checks.
Fill in the Blanks
Cloze items for vocabulary and method.
Matching
Pair terms, models and definitions.
Short Questions
One to three mark responses.
Long Questions
Extended responses with method marks.
Exam Questions
Board-styled questions and command words.
Past Papers
Authentic papers filtered to this entity.
Solved Papers
Full worked solutions and examiner notes.
Frequently asked
- What sits underneath Counterexamples in the knowledge graph?
- 0 domains, 0 strands, 0 units, 3 chapters, 12 topics, 60 subtopics, 360 concepts and 1,800 micro-concepts — every one of them carrying its own worksheets, quizzes, flashcards, lesson plans and AI tutoring.
- What should a learner already know before Counterexamples?
- The graph lists Matrices as prerequisites. Each links to its own practice so a gap can be closed before moving on.
- Which grades is Counterexamples taught in?
- Typically Grade 3, Grade 4, Grade 5, though every resource can be regenerated for any grade in the platform.
Related resources
Work through Counterexamples one-to-one with a tutor who can see exactly where the gap is.
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